# Questions tagged [calculus-of-variations]

This tag is for problems relating to the calculus of variations that deal with maximizing or minimizing functionals. This problem is a generalization of the problem of finding extrema of functions of several variables. In fact, these variables will themselves be functions and we will be finding extrema of “functions of functions” or functionals.

2,600 questions
Filter by
Sorted by
Tagged with
11 views

### How to write the Euler-Langrange equation of elliptic PDE containing $\nabla$ like $\Delta u + \nabla u . \theta = f$.

How to write the Euler-Langrange equation of elliptic PDE containing $\nabla$ like $\Delta u + \nabla u . \theta = f$. For some question we can make the $\nabla$ term dissapear by transformation, then ...
15 views

71 views

### Free boundary geodesics as a critical point of the energy functional

As a consequence of the formula for the first variation of the energy of a curve, we have the following known characterization of geodesics. A piecewise differentiable curve $c:[0,1]\to M$ is a ...
29 views

### Gradient of a functional defined on an Hilbert space (with respect to a $W^{1,2}$ inner product)

$\newcommand{\R}{\mathbb R}$ Consider the Hilbert space $X = W^{1,2}(\R)\oplus W^{1,2}(\R)$ (Sobolev spaces). I define a function $F:X\to \R$ as $$F(u,g)= \int_\R u(t)\partial_tg(t) dt.$$ $F$ is ...
51 views

36 views

### Extremals of $\int_0^\pi(y')^2dx$

I am trying to show that $\int_0^\pi(y')^2dx$ with $y(0) = 0$, $y(\pi) = 0$ has infinitely many extremals subject to the constraint $\int_0^\pi y^2 dx = \pi/2$. I know that the first-order necessary ...
27 views

### Can we characterize functions, like exponentials, the gamma function, and tetration, as solutions of an optimization problem?

This is something I recently started wondering about. I've long been interested in the idea of problems of the form "given a sequence of real numbers $a_n$, under what cases is there some way to ...
1 vote
50 views

### Texts about global minima in functionals

I am doing some numerics where I found the minimal value of a functional that does not satisfy the Euler-Lagrange equation associated. I think I am dealing with a minimal value that is not a local ...
17 views

91 views

### Formal definition of variation

I'm trying to read a book on classical mechanics, and I'm having a hard time trying to know what exactly is a variation. In the Lagrangian "formalism" the differential forms $dx$ are changed ...
1 vote
45 views

77 views

### Along what curve does a circle roll down the fastest?

In this MO question, I stated that a circle rolls down the fastest along a cycloid curve - as shown by Johan Bernoulli, thereby solving the Brachistochrone problem. However, as user Manfred Weis ...
9 views

### Euler's equation different boundary conditions

I am trying to find the extremal of a functional I, the boundary conditions on the integral are [0,π/6] but the boundary conditions are y(0) = 0 and y(π) = 2, I am confused about why the boundary ...
36 views

### Finding the path of a light ray using differential geometry

Hi I am trying to solve a calculus of variations problem I would like to solve it using differential geometric approach but i am not sure. As an example how would I go back doing it for the example ...
27 views

29 views

### What $is$ $\text{dist}^2\left( F,\mathsf{SO}(3)\right)$?

Reading some review about calculous of variations and mechanics, the following notation is very recurrent for energy density functionals \begin{equation} \text{dist}^p\left( F,\mathsf{SO}(N)\right), \...
1 vote
I'm interested in minimizing the functional $I[f]=\int_{a}^{b} \mathcal{L}(x,f,f',f'',f^{-1},(f^{-1})', (f^{-1})'') dx$. Would it be allowable to consider that $f^{-1}$ is some function $g$ and apply ...
### Capturing $\infty$ endpoint and additional boundary conditions
I'm interested in minimizing the functional $I[f]=\int_{a}^{\infty} J(x,f(x),f'(x)) dx$ but the boundary conditions that I have on $f$ are a bit "weird": one endpoint is fixed in the ...